Having been 13-0 down just past the half-hour mark, the Portuguese fought back with 18 unanswered points before Tengizi Zamtaradze’s 78th-minute try, given after a review, drew things level.
Luka Matkava had the chance to put Georgia in front but could not make the conversion from a tough angle, and Portugal were then awarded a last-gasp penalty – only for Nuno Sousa Guedes to drag his kick wide.
Georgia had opened the scoring with Akaki Tabutsadze’s second-minute try, converted by Tedo Abzhandadze, who added two penalties.
Portugal then got off the mark from their first real opportunity, with Raffaele Storti crossing in the 34th minute.
Samuel Marques was unable to convert on that occasion, hitting the upright, but was successful with two penalties early in the second half.
Storti then scored his second try in the 57th minute to take his side into the lead, with Marques adding the extras, before the compelling contest entered its thrilling finale.
Georgia, fourth in Pool C, will next face Fiji in Bordeaux on Saturday, while fifth-placed Portugal – for whom this was a first match at this level avoiding defeat, in their second World Cup campaign – take on Australia in St Etienne the next day.
Additional reporting by PA Sport.
